Euclid's Parallel Postulate Proof

Question

I was checking this page: http://math2.org/math/misc/unproved.htm

And Euclid's Parallel Postulate caught my interest. However, I don't fully seem to understand it... What are we required to prove? That the orange & blue lines are parallel or what?

Image

Or do we have to prove that only a single line (orange one) is parallel to the blue line?

Answer 1

This can't be proven. Many people in the past have tried to prove that it's true that (1) there always exists a line parallel to another line through a point and that (2) it is unique. The only meaningful proof would be something axiomatic, using the other four of Euclid's postulates:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

4. All right angles are congruent.

However, the parallel postulate has been shown to be independent of these. If you include it as an axiom, you get "Euclidean Geometry" which is basically all of the geometry you learn until the end of high school. If you do not include the parallel postulate (or use other axioms, like "every line intersects every other line exactly once") you end up with many other "geometries". You should look into hyperbolic and elliptic geometry!